There's actually not. Unless it infringes on a protected class (eg "no black people") or a fundamental liberty interest, the "rational basis" standard prevails. This basically means that if a hypothetical reason could be articulated to support a public interest it's legal, even if it does not actually achieve that goal in practice.
This becomes particularly broad when you're talking about actions in public. So while it would probably be considered an intrusion to send the Dental Police into everyone's home to be sure that they floss daily, it would probably be legal to have the Dental Police perform mandatory tooth-brushings on all taxi drivers. Nobody is forcing you to be a taxi driver and you could reasonably avoid the situation by practicing another profession instead.
That's how you get stuff like requiring hair nets for food safety. But again, it doesn't necessarily have to be sensical. Unless otherwise restricted, states and cities have an enormous amount of leeway to do whatever they think is in their citizens' best interests. It's a list of things you can't do, not a list of things you can.
If Austin wants to fingerprint taxi drivers, that's well within their powers. Not even controversial - that has been common practice for probably a half century now.